Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 261, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY OF
POSITIVE PERIODIC SOLUTIONS OF A LOTKA-VOLTERRA
TYPE COMPETITION SYSTEMS WITH DELAYS AND
FEEDBACK CONTROLS
ANH TUAN TRINH
Abstract. The existence of positive periodic solutions of a periodic LotkaVolterra type competition system with delays and feedback controls is studied
by applying the continuation theorem of coincidence degree theory. By contracting a suitable Liapunov functional, a set of sufficient conditions for the
global asymptotic stability of the positive periodic solution of the system is
given. A counterexample is given to show that the result on the existence of
positive periodic solution in [4] is incorrect.
1. Introduction
In this paper, we consider the following non-autonomous Lotka-Volterra n-species
competition system with delays and feedback controls
n
n X
m
h
X
X
ẋi (t) = gi (xi (t)) ri (t) −
aij (t)xj (t) −
bijk (t)xj (t − τijk (t))
j=1
−
−
n X
m Z t
X
j=1 k=1
cijk (t, s)xj (s)ds − di (t)ui (t)
−∞
j=1 k=1
m
X
Z
t
eik (t)ui (t − σik (t)) −
i
fi (t, s)ui (s)ds ,
(1.1)
−∞
k=1
u̇i (t) = −αi (t)ui (t) + βi (t)xi (t) +
m
X
pik (t)xi (t − γik (t))
k=1
Z
t
+
vi (t, s)xi (s)ds,
−∞
where i ∈ {1, 2, . . . , n}, ui denote indirect feedback control variables. For system
(1.1), we introduce the following hypotheses
2000 Mathematics Subject Classification. 34K13, 92B05, 92D25, 93B52.
Key words and phrases. Lotka-Volterra competition system; time delay; feedback control;
positive periodic solution; global asymptotic stability; continuation theorem.
c
2013
Texas State University - San Marcos.
Submitted January 8, 2013. Published November 26, 2013.
1
2
A. T. TRINH
EJDE-2013/261
(H1) ri , αi ∈ C(R, R), aij , bijk , diR, eik , pik , βi ∈ C(R,
R ω R+ ) are ω-periodic (ω is a
ω
fixed positive number) with 0 ri (t)dt > 0, 0 αi (t)dt > 0, i, j = 1, 2, . . . , n;
k = 1, 2, . . . , m.
(H2) cijk , fi , vi : R × R → R+ are ω-periodic functions; i.e.,
cijk (t + ω, s + ω) = cijk (t, s), fi (t + ω, s + ω) = fi (t, s), vi (t + ω, s + ω) = vi (t, s)
Rt
Rt
Rt
and −∞ cijk (t, s)ds, −∞ fi (t, s)ds, −∞ vi (t, s)ds are continuous with respect to t. Moreover
+∞
Z
0
Z
Z
+∞
Z
0
fi (s + t, s) ds dt < +∞,
cijk (s + t, s) ds dt < +∞,
−t
0
Z
+∞
−t
0
0
Z
vi (s + t, s) ds dt < +∞,
i, j = 1, 2, . . . , n; k = 1, 2, . . . , m.
−t
0
(H3) gi ∈ C(R+ , R+ ) is strictly increasing, gi (0) = 0 and limv→0+ giv(v) is a
positive constant. Moreover, there are positive constants l and L such that
l ≤ giv(v) ≤ L for all v > 0, i = 1, 2, . . . , n.
(H4) τijk , σik , γik ∈ C(R, R+ ) are ω-periodic for i, j = 1, 2, . . . n, k = 1, 2, . . . , m.
(H5) τijk , σik , γik ∈ C 1 (R, R+ ) and τ̇ijk (t) < 1, σ̇ik (t) < 1, γ̇ik (t) < 1 for all
t ∈ R, i, j = 1, 2, . . . , n, k = 1, 2, . . . , m.
We consider (1.1) with the initial conditions
xi (s) = φi (s), s ∈ (−∞, 0],
ui (s) = ψi (s), s ∈ (−∞, 0],
φi ∈ C((−∞, 0], R+ ), φi (0) > 0
ψi ∈ C((−∞, 0], R+ ),
ψi (0) > 0,
(1.2)
for i = 1, 2, . . . , n. Throughout this paper, we use the following symbols: for an
ω-periodic function f ∈ C(R, R), we define
1
f¯ =
ω
Z
ω
f (t)dt,
c∗ijk (t) =
Z
t
cijk (t, s)ds,
−∞
0
fi∗ (t) =
Z
t
fi (t, s)ds,
−∞
Rs
exp{ t αi (v)dv}
Rω
, s ≥ t,
=
vi (t, s)ds, Gi (t, s) =
exp{ 0 αi (v)dv} − 1
−∞
Z t+ω
m
h
i
X
Pi (t) = di (t)
Gi (t, s) βi (s) +
pik (s) + vi∗ (s) ds,
Z
vi∗ (t)
t
t
Qi (t) =
m
X
t
j=1
Z
t
Ri (t) =
Z
fi (t, s)
−∞
t+ω
Z
eij (t)
s
s+ω
k=1
m
X
h
Gi (t, s) βi (s) +
(1.3)
i
pik (s) + vi∗ (s) ds,
k=1
m
h
i
X
Gi (s, τ ) βi (τ ) +
pik (τ ) + vi∗ (τ ) dτ ds.
k=1
Fan et al. [1] studied system (1.1) in the case k = 1, gi (vi ) = vi , fi (t, s) =
vi (t, s) = 0 for i = 1, 2, . . . , n and obtained several results for the existence and
global asymptotically stability of positive periodic solutions of the system. Recently,
Yan and Liu [4] considered system (1.1) in the case eik (t) = pik (t) = 0 and fi (t, s) =
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EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY
3
vi (t, s) = 0 for k = 2, 3, . . . , m, i = 1, 2, . . . , n,
n
n X
m
h
X
X
ẋi (t) = gi (xi (t)) ri (t) −
aij (t)xj (t) −
bijk (t)xj (t − τijk (t))
j=1
−
n X
m Z
X
j=1 k=1
t
j=1 k=1
i (1.4)
cijk (t, s)xj (s)ds − di (t)ui (t) − ei1 (t)ui (t − σi1 (t)) ,
−∞
u̇i (t) = −αi (t)ui (t) + βi (t)xi (t) + pi1 (t)xi (t − γi1 (t)),
i = 1, 2, . . . , n.
By employing fixed point index theory on cones, Yan and Liu [4] established the
following result.
Theorem 1.1 ([4]). Assume that (H1)–(H4) hold. For system (1.4), to have at
least one positive ω-periodic solution, a necessary and sufficient condition is
n h
m
i
nX
o
X
āij +
(b̄ijk + c̄∗ijk ) + P̄i + Q̄i > 0
min
(1.5)
1≤i6n
j=1
k=1
and
min {β̄i + p̄i1 } > 0.
(1.6)
1≤i≤n
Unfortunately, the sufficient condition in the above theorem is incorrect, as
shown by the following example.
Example. Consider the system
h
i
ẋ1 (t) = x1 (t) 1 − x1 (t) − 3x2 (t) − 2x2 (t − τ1 ) − u1 (t) − u1 (t − σ1 ) ,
h
i
(1.7)
ẋ2 (t) = x1 (t) 2 − 3x1 (t) − x2 (t) − 2x1 (t − τ2 ) − u2 (t) − u2 (t − σ2 ) ,
u̇1 (t) = −u1 (t) + x1 (t) + x1 (t − γ1 ), u̇2 (t) = −u2 (t) + x2 (t) + x2 (t − γ2 ),
where τi , σi , γi , i = 1, 2, are positive constants. It is easy to see that system
(1.7) satisfies all hypotheses of Theorem 1.1 with ω = 1. On the other hand, if
(x∗1 (t), x∗2 (t), u∗1 (t), u∗2 (t)) is a positive 1-periodic solution of system (1.7), then
h
i
d
ln x∗1 (t) = 1 − x∗1 (t) − 3x∗2 (t) − 2x∗2 (t − τ1 ) − u∗1 (t) − u∗1 (t − σ1 ) ,
dt
h
i
d
ln x∗2 (t) = 2 − 3x∗1 (t) − x∗2 (t) − 2x∗1 (t − τ2 ) − u∗1 (t) − u∗2 (t − σ2 ) ,
dt
(1.8)
d ∗
u1 (t) = −u∗1 (t) + x∗1 (t) + x∗1 (t − γ1 ),
dt
d ∗
u (t) = −u∗2 (t) + x∗2 (t) + x∗2 (t − γ2 ),
dt 2
Integrating (1.8) from 0 to 1 and simplifying, we obtain
x̄∗1 + 5x̄∗2 + 2ū∗1 = 1,
5x̄∗1 + x̄∗2 + 2ū∗2 = 2,
ū∗1 = 2x̄∗1 ,
ū∗2 = 2x∗2 .
(1.9)
This implies
5x̄∗1 + 5x̄∗2 = 1, 5x̄∗1 + 5x̄∗2 = 2,
which is impossible. Thus, system (1.7) has no positive 1-periodic solution and
then the sufficient condition in Theorem 1.1 (given in [4]) is incorrect.
In the proof of Theorem 1.1 (see [4]), authors considered a map Φ : K → K,
where
K = {(x1 , . . . , xn ) ∈ E : xi (t) > δi kxi k0 , i = 1, 2, . . . , n, t ∈ [0, ω]}
4
A. T. TRINH
EJDE-2013/261
n
(with δi = exp{−r̄i ω}) is a cone of the Banach space
Pn E = {x ∈ C(R, R ) :
x(t + ω) = x(t) for all t ∈ R} with the norm kxk0 = i=1 kxi k0 (where kxi k0 =
maxt∈[0,ω] |xi (t)|). By employing fixed point index theory on cone, it was proved
in [4] that there exist positive constants r and R (r < R) such that Φ has at least
one fixed point x∗ in Kr,R = {x ∈ K : r < kxk0 ≤ R}, and then it was concluded
that (x∗ (t), u∗ (t)) is positive ω-periodic solution of system (1.1). The mistake in
the proof of Theorem 1.1 (see [4]) is that x∗ ∈ Kr,R does not imply that x∗ (t) is
positive for n ≥ 2.
Our purpose of this paper is by using the technique of coincidence degree theory
developed by Gains and Mawhin in [2] to study the existence of positive periodic
solutions of system (1.1). Moreover, by contracting a suitable Liapunov functional,
we study the global asymptotic stability of the positive periodic solution of system
(1.1). The remainder of this paper is organized as follows. Section 2 is preliminaries, in which we introduce the continuation theorem and some lemmas. Section 3
contains our main results on the existence and the global asymptotic stability of
positive periodic solutions of system (1.1).
2. Preliminaries
Let Y and Z be two normed vector spaces, L : Dom L ⊂ Y → Z be a linear
mapping, and N : Y → Z be a continuous mapping. The mapping L will be called a
Fredholm mapping of index zero if dim ker L = codim Im L < +∞ and Im L is closed
in Z. If L is a Fredholm mapping of index zero and there exist continuous projectors
P : Y → Y and Q : Z → Z such that Im P = ker L, Im L = ker Q = Im(I − Q), it
follows that L|Dom L∩ker P : (I − P )X → Im L is invertible. We denote the inverse
of that map by KP . If Ω is an open bounded subset of Y , the mapping N will be
called L-compact on Ω̄ if QN (Ω̄) is bounded and KP (I − Q)N : Ω̄ → Y is compact.
Since Im Q is isomorphic to ker L, there exists an isomorphism J : Im Q → ker L.
Lemma 2.1 (Continuation theorem [2]). Let L be a Fredholm mapping of index
zero and N be L-compact on Ω̄. Suppose that
(i) for each λ ∈ (0, 1), every solution x of Lx = λN x is such that x 6∈ ∂Ω;
(ii) QN x 6= 0 for each x ∈ ∂Ω ∩ ker L and deg(JQN, Ω ∩ ker L, 0) 6= 0.
Then the operator equation Lx = N x has at least one solution lying in Dom L ∩ Ω̄.
Let us consider the system
h
ẋi (t) = gi (xi (t)) ri (t) −
n
X
aij (t)xj (t) −
j=1
−
−
n X
m Z
X
j=1 k=1
m
X
n X
m
X
bijk (t)xj (t − τijk (t))
j=1 k=1
t
cijk (t, s)xj (s)ds − di (t)(Vi xi )(t)
−∞
Z
t
eik (t)(Vi xi )(t − σik (t)) −
i
fi (t, s)(Vi xi )(s)ds ,
−∞
k=1
ui (t) = (Vi xi )(t),
i = 1, 2, . . . , n,
(2.1)
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EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY
5
where
Z
t+ω
(Vi xi )(t) =
m
h
X
Gi (t, s) βi (s)xi (s) +
pik (s)xi (s − γik (s))
t
k=1
Z
s
+
i
vi (s, τ )xi (τ )dτ ds,
(2.2)
i = 1, 2, . . . , n.
−∞
Lemma 2.2. Assume (H1)–(H4) hold. Then (x1 (t), . . . , xn (t), u1 (t), . . . , un (t)) is
an ω-periodic solution of system (1.1) if and only if it is an ω-periodic solution of
system (2.1).
Proof. (⇒) Let (x(t), u(t)) = (x1 (t), . . . , xn (t), u1 (t), . . . , un (t)) be an ω-periodic
solution of system (1.1). It is easy to see from (1.1) that
Z t̄
Z t̄
Z τ
h
n
ui (t̄) = exp{−
αi (τ )dτ } ui (t) +
exp{
αi (τ )dτ } βi (τ )xi (τ )
t
+
m
X
t
Z
t
τ
pik (τ )xi (t − γik (τ )) +
o i
vi (τ, v)xi (v)dv dτ ,
−∞
k=1
for t̄ > t, i = 1, 2, . . . , n. Thus, since ui (t) = ui (t + ω) for i = 1, 2, . . . , n, it follows
that
ui (t) = ui (t + ω)
Z t+ω
Z
h
= exp{−
αi (τ )dτ } ui (t) +
t
+
m
X
t+ω
Z
exp{
t
Z
τ
pik (τ )xi (t − γik (τ )) +
τ
n
αi (τ )dτ } βi (τ )xi (τ )
t
o i
vi (τ, v)xi (v)dv dτ ,
−∞
k=1
for t ∈ R, i = 1, 2, . . . , n. Hence,
Z t+ω
m
h
X
ui (t) =
pik (τ )xi (τ − γik (τ ))
Gi (t, τ ) βi (τ )xi (τ ) +
t
k=1
Z
s
i
+
vi (τ, v)xi (v)dv dτ = (Vi xi )(t),
i = 1, 2, . . . , n,
−∞
which implies that (x(t), u(t)) is an ω-periodic solution of system (2.1).
(⇐) Let (x1 (t), . . . , xn (t), u1 (t), . . . , un (t)) be an ω-periodic solution of system
(2.1). It is easy to see from (2.2) that (u1 (t), . . . , un (t)) satisfies the system
Z t
m
X
u̇i (t) = −αi (t)ui (t) + βi (t)xi (t) +
pik (t)xi (t − γik (t)) +
vi (t, s)xi (s)ds,
k=1
−∞
for i = 1, 2, . . . , n. Thus, (x(t), u(t)) is an ω-periodic solution of system (1.1). The
proof is complete.
Lemma 2.3 ([3]). Suppose that ν ∈ C 1 (R, R+ ) is ω-periodic and ν 0 (t) < 1 for all
t ∈ [0, ω]. Then the function t−ν(t) has a unique inverse η(t) satisfying η ∈ C(R, R)
with η(t + ω) = η(t) for all t ∈ R.
Lemma 2.4. Suppose that qij > 0, δi > 0 for i, j = 1, 2, . . . , n. If
δi −
n
X
j6=i
qij
δj
> 0, i = 1, 2, . . . , n,
qjj
(2.3)
6
A. T. TRINH
then the system of algebraic equations
n
X
δi −
qij xj = 0,
EJDE-2013/261
i = 1, 2, . . . , n
(2.4)
j=1
has a unique solution x∗ = (x∗1 , . . . , x∗n ) ∈ Rn . Moreover, x∗i > 0 for i = 1, 2, . . . , n.
Proof. Let yi = qii xi for i = 1, 2, . . . , n, so that system (2.4) becomes
n
X
∗
δi −
qij
yj = 0, i = 1, 2, . . . , n,
(2.5)
j=1
∗
∗
where qij
= qij /qjj , i, j = 1, 2, . . . , n. Clearly, qii
= 1 for i = 1, 2, . . . , n. By (2.3),
δi −
n
X
∗
qij
δj > 0,
i = 1, 2, . . . , n.
(2.6)
j6=i
Pn ∗ Let be a positive number such that < min1≤i≤n δi − j6=i qij
δj . Denote
D = {y = (y1 , . . . , yn ) ∈ Rn : ≤ yi ≤ δi , i = 1, 2, . . . , n},
F = (F1 , . . . , Fn ) : Rn → Rn , where
n
X
∗
Fi (y) = δi −
qij
yj ,
i = 1, 2, . . . , n;
j=1
and H = F + I, where I is the identity operator on Rn . It is easy to see that
< Hi (y) < δi for i = 1, 2, . . . , n, y ∈ D; i.e., H(y) ∈ int(D)-the interior of D for
all y ∈ D. Thus, deg(H − I, int D, 0) = deg(F, int D, 0) = 1. This implies that the
equation F y = 0 has at least one solution y ∗ in int D. Since F y 6= 0 for all y ∈ ∂D
and F y = 0 is linear equation, it follows that y ∗ is the unique solution in Rn of the
equation F y = 0. Thus, equation (2.4) has a unique solution x∗ = (x∗1 , . . . , x∗n ) ∈
Rn . Moreover, x∗i > 0 for i = 1, 2, . . . , n. The proof is complete.
Definition 2.5. Let (x̃(t), ũ(t)) = (x̃1 (t), . . . , x̃n (t), ũ1 (t), . . . , ũn (t)) be a positive
ω-periodic solution of system (1.1). It is said to be globally asymptotically stable if
any positive solution (x(t), u(t)) = (x1 (t), . . . , xn (t), u1 (t), . . . , un (t)) of (1.1)-(1.2)
satisfies
n n
o
X
lim
|xi (t) − x̃i (t)| + |ui (t) − ũi (t)| = 0.
t→+∞
i=1
Rx
Remark 2.6. Let us put yi = hi (xi ) := 1 i gids
(s) , i = 1, 2, . . . , n. By (H3), it
is easy to see that hi : (0, +∞) → R, xi 7→ yi = hi (xi ) has a unique inverse
ϕi : R → (0, +∞), yi 7→ xi = ϕi (yi ). Moreover, ϕi ∈ C 1 (R, (0, +∞)) and ϕi is
strictly monotone increasing.
Remark 2.7. By (H5), Lemma 2.3 implies that the functions t − τijk (t), t − σik (t)
and t − γik (t) have the unique inverses, respectively. Let µijk (t), ζik (t) and ξik (t)
represent the inverses of functions t − τijk (t), t − σik (t) and t − γik (t), respectively. Obviously, µijk , ζik , ξik ∈ C(R, R) and µijk (t + ω) = µijk (t), ζik (t + ω) =
ζik (t), ξik (t + ω) = ξik (t) for all t ∈ R.
Remark 2.8. It is easy from (H1)–(H4) to show that solutions of (1.1)-(1.2) are
well defined for all t ≥ 0 and satisfy xi (t) > 0 and ui (t) > 0 for all t ≥ 0 and
i = 1, 2, . . . , n.
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EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY
7
3. Main Results
Theorem 3.1. Assume that (H1)–(H4) hold. Let
Z
ω
h
βi (s) +
0
Ai := āii +
r̄i >
n X
āij +
j6=i
m
X
m
X
i
pik (s) + vi∗ (s) ds > 0,
k=1
m
X
c̄∗iik + P̄i + Q̄i + R̄i > 0,
b̄iik +
k=1
m
X
k=1
m
X
k=1
k=1
b̄ijk +
i = 1, 2, . . . , n,
i = 1, 2, . . . , n,
c̄∗ijk + P̄j + Q̄j + R̄j ϕj (Bj ),
(3.1)
(3.2)
i = 1, 2, . . . , n, (3.3)
where Bi = hi (r̄i /Ai ) + (r̄i + |ri |)ω, i = 1, 2, . . . , n. Then system (1.1) has at least
one positive ω-periodic solution.
Proof. Consider the system
ẏi (t) = ri (t) −
n
X
aij (t)ϕj (yj (t)) −
j=1
−
−
m Z
n X
X
j=1 k=1
m
X
m
n X
X
bijk (t)ϕj (yj (t − τijk (t)))
j=1 k=1
t
cijk (t, s)ϕj (yj (s))ds − di (t)(Vi ϕi (yi ))(t)
(3.4)
−∞
Z
t
eik (t)(Vi ϕi (yi ))(t − σik (t)) −
fi (t, s)(Vi ϕi (yi ))(s)ds.
−∞
k=1
By (1.3), (2.2) and (3.1), if system (3.4) has an ω-periodic solution (y1∗ (t), . . . , yn∗ (t)),
then
u∗i (t) = (Vi ϕi (yi∗ ))(t)
Z t+ω h
1
>
min ϕi (yi∗ (τ )) βi (s)
exp{ᾱi ω} − 1 τ ∈[0,ω]
t
m
i
X
+
pik (s) + vi∗ (s) ds > 0, t ∈ R, i = 1, 2, . . . , n.
k=1
Thus, (x∗1 (t), . . . , x∗n (t), u∗1 (t), . . . , u∗n (t)) with values x∗i (t) = ϕi (yi∗ (t)) and u∗i (t) =
(Vi ϕi (yi∗ ))(t) for i = 1, 2, . . . , n is a positive ω-periodic solution of system (2.1). So,
by Lemma 2.2, we only need to show that system (3.4) has at least one ω-periodic
solution in order to complete the proof. To apply the continuation theorem of
coincidence degree theory to the existence of an ω-periodic solution of system (3.4),
we take
n
o
Y = Z = y(t) = (y1 (t), . . . , yn (t)) ∈ C(R, Rn ) : y(t + ω) = y(t) for all t ∈ R .
Pn
Denote kyk0 = i=1 kyi k0 , where kyi k0 = maxt∈[0,ω] |yi (t)|. Then Y and Z are
Banach spaces when they endowed with the norm k · k0 .
8
A. T. TRINH
EJDE-2013/261
We define L : Dom L ⊂ Y → Z and N : Y → Z by setting Ly = ẏ and
N y = F y = (F1 y, . . . , Fn y), where
Fi y(t) =ri (t) −
−
−
n
X
aij (t)ϕj (yj (t)) −
j=1
n X
m Z t
X
j=1 k=1
m
X
n X
m
X
bijk (t)ϕj (yj (t − τijk (t)))
j=1 k=1
cijk (t, s)ϕj (yj (s))ds − di (t)(Vi ϕi (yi ))(t)
(3.5)
−∞
Z
t
eik (t)(Vi ϕi (yi ))(t − σik (t)) −
fi (t, s)(Vi ϕi (yi ))(s)ds.
−∞
k=1
Further, we define continuous projectors P : Y → Y and Q : Z → Z as follows
Z
Z
1 ω
1 ω
y(s)ds, Qz =
z(s)ds.
Py =
ω 0
ω 0
Rω
We easily see that Im L = {z ∈ Z : 0 z(s)ds = 0} and ker L = Rn . So, Im L is
closed in Z and dim ker L = n = codim Im L. Hence, L is a Fredholm mapping of
index zero. Clearly that Im P = ker L, Im L = ker Q = Im(I − Q)/ Furthermore,
the generalized inverse (to L) KP : Im L → ker P ∩ Dom L has the form
Z t
Z Z
1 ω t
KP z(t) =
z(s)ds −
z(s) ds dt.
ω 0 0
0
We know that
1
QN y(t) =
ω
Z
ω
F y(t)dt.
0
Thus,
KP (I − Q)N y(t) = (KP N − KP QN )y(t)
Z
Z t
Z Z
1
t ω
1 ω t
=
−
F y(s)ds.
F y(s)ds −
F y(s) ds dt +
ω 0 0
2 ω 0
0
It is easy to see that QN and Kp (I − Q)N are continuous. Furthermore, it can
be verified that KP (I − Q)N (Ω̄) is compact for any open bounded set Ω ⊂ Y by
using Arzela-Ascoli theorem and QN (Ω̄) is bounded. Therefore, N is L-compact
on Ω̄ for any open bounded subset Ω ⊂ Y . Now we are in a position to search
for an appropriate open bounded subset Ω for the application of the continuation
theorem (Lemma 2.1) to system (3.4).
Corresponding to the operator equation Ly = λN y (λ ∈ (0, 1)), we have
n
n X
m
h
X
X
ẏi (t) = λ ri (t) −
aij (t)ϕj (yj (t)) −
bijk (t)ϕj (yj (t − τijk (t)))
−
−
j=1
Z
n
m
XX t
j=1 k=1
m
X
j=1 k=1
cijk (t, s)ϕj (yj (s))ds − di (t)(Vi ϕi (yi ))(t)
−∞
Z
t
eik (t)(Vi ϕi (yi ))(t − σik (t)) −
k=1
−∞
i
fi (t, s)(Vi ϕi (yi ))(s)ds .
(3.6)
EJDE-2013/261
EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY
9
Integrating (3.6) from 0 to ω and simplifying, we obtain
n Z ω
n X
m Z ω
X
X
r̄i ω =
aij (t)ϕj (yj (t))dt +
bijk (t)ϕj (yj (t − τijk (t)))dt
j=1 0
n X
m Z ω
X
j=1 k=1
Z
Z
+
+
0
t
cijk (t, s)ϕj (yj (s)) ds dt +
−∞
j=1 k=1 0
m Z ω
X
ω
di (t)(Vi ϕi (yi ))(t)dt
0
(3.7)
eik (t)(Vi ϕi (yi ))(t − σik (t))dt
k=1 0
Z ωZ t
fi (t, s)(Vi ϕi (yi ))(s) ds dt.
+
0
−∞
Let
yi (ηi ) = max yi (t), yi (θi ) = min yi (t),
t∈[0,ω]
t∈[0,ω]
ηi , θi ∈ [0, ω], i = 1, 2, . . . , n.
(3.8)
It is easy to see from (1.3), (2.2) and (3.8) that
Z ω
di (t)(Vi ϕi (yi ))(t)dt ≥ P̄i ωϕi (yi (θi )),
0
m Z
X
ω
eik (t)(Vi ϕi (yi ))(t − σik (t))dt ≥ Q̄i ωϕi (yi (θi )),
(3.9)
0
k=1
ω
Z
t
Z
fi (t, s)(Vi ϕi (yi ))(s) ds dt ≥ R̄i ωϕi (yi (θi ))
−∞
0
and
ω
Z
di (t)(Vi ϕi (yi ))(t)dt ≤ P̄i ωϕi (yi (ηi )),
0
m Z
X
eik (t)(Vi ϕi (yi ))(t − σik (t))dt ≤ Q̄i ωϕi (yi (ηi )),
(3.10)
0
k=1
Z
ω
ω
t
Z
fi (t, s)(Vi ϕi (yi ))(s) ds dt ≤ R̄i ωϕi (yi (ηi )).
−∞
0
It follows from (1.3), (3.7), (3.8) and (3.9) that
m
m
X
X
r̄i ≥ āii +
b̄iik +
c̄∗iik + P̄i + Q̄i + R̄i ϕi (yi (θi )) = Ai ϕi (yi (θi )),
k=1
(3.11)
k=1
for i = 1, 2, . . . , n. Thus, by (3.2) and Remark 2.6, we have
r̄i
yi (θi ) ≤ hi ( ), i = 1, 2, . . . , n.
Ai
From (3.6) and (3.7), we know that
Z ω
|ẏi (t)|dt ≤ (r̄i + |ri |)ω,
(3.12)
i = 1, 2, . . . , n,
(3.13)
0
and thus, by (3.12),
Z
yi (t) ≤ yi (θi ) +
0
ω
|ẏi (t)|dt ≤ hi (
r̄i
) + (r̄i + |ri |)ω = Bi ,
Ai
t ∈ [0, ω],
(3.14)
10
A. T. TRINH
EJDE-2013/261
for i = 1, 2, . . . , n. It is easy to see from (3.7), (3.8), (3.10) and (3.14) that
āii +
m
X
b̄iik +
k=1
n X
> r̄i −
m
X
c̄∗iik + P̄i + Q̄i + R̄i ϕi (yi (ηi ))
k=1
m
X
āij +
j6=i
b̄ijk +
k=1
m
X
(3.15)
c̄∗ijk
+ P̄j + Q̄j + R̄j ϕj (yj (ηj )),
k=1
for i = 1, 2, . . . , n. Thus, by (3.3) and (3.14), we have
Pn Pm
r̄i − j6=i āij + k=1 [b̄ijk + c̄∗ijk ] + P̄j + Q̄j + R̄j ϕj (Bj )
Pm
=: Ci ,
ϕi (yi (ηi )) ≥
āii + k=1 [b̄iik + c̄∗iik ] + P̄i + Q̄i + R̄i
for i = 1, 2, . . . , n; or
yi (ηi ) ≥ hi (Ci ),
i = 1, 2, . . . , n,
From (3.13) and (3.16), it follows that
Z ω
y(t) ≥ yi (ηi ) −
|ẏi (t)|dt ≥ hi (Ci ) − (r̄i + |ri |)ω =: Di , t ∈ [0, ω],
(3.16)
(3.17)
0
for i = 1, 2, . . . , n. From (3.12) and (3.16) we see that
kyk0 ≤ M :=
n
X
(|Bi | + |Di |).
(3.18)
i=1
By (3.3), Lemma 2.3 implies that the following system of algebraic equations
r̄i =
n h
X
āij +
j=1
m
X
b̄ijk +
k=1
m
X
i
c̄∗ijk ϕj (yj ) + (P̄i + Q̄i + R̄i )ϕi (yi ),
(3.19)
k=1
for i = 1, 2, . . . , n, has a unique solution y ∗ = (y1∗ , . . . , yn∗ ) ∈ Rn . Let S = ky ∗ k0 +M .
Evidently, S is independent of the choice of λ. Let Ω := {y ∈ Y : kyk1 < S}. It
is clear that Ω satisfies the requirement (i) in Lemma 2.1. Moreover, QN y 6= 0 for
any y ∈ ∂Ω ∩ ker L. Let us take J = I, where I is the identity operator on Rn . By
straightforward computation, we have
n
h
i
o
deg(JQN, Ω ∩ ker L, 0) = sgn (−1)n det(wij ) ϕ1 (y1∗ ) . . . ϕn (yn∗ ) 6= 0,
where wii = Ai ,
wij = āij +
m
X
k=1
b̄ijk +
m
X
c̄∗ijk (j 6= i),
i, j = 1, 2, . . . , n.
k=1
By Lemma 2.1, we conclude that the equation Ly = N y has at least one solution
in Y . Therefore, system (1.1) has at least one positive ω-periodic solution. The
proof is complete.
EJDE-2013/261
EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY
11
Theorem 3.2. Assume that (H1)–(H5), (3.1) and (3.3) hold. If there exist positive
constants ν1 , ν2 , . . . , νn such that
min
t∈[0,ω]
−
n
X
m
n h
X
νi aii (t) − βi (t) −
k=1
νj aji (t) −
−
j=1
min
t∈[0,ω]
νj
j=1
j6=i
n
X
n
X
νj
m Z
X
m
X
pik (ξik (t))
−
1 − γ̇ik (ξik (t))
+∞
Z
vi (t + τ, t)dτ
i
0
bjik (µjik (t))
1 − τ̇jik (µjik (t))
k=1
(3.20)
+∞
cjik (t + τ, t)dτ
o
> 0,
0
k=1
m
n
X
αi (t) − di (t) −
k=1
eik (ζik (t))
−
1 − σ̇ik (ζik (t))
Z
+∞
fi (t + τ, t)dτ
o
> 0,
0
for i = 1, 2, . . . , n, then system (1.1) has a unique positive ω-periodic solution which
is globally asymptotically stable.
Proof. From (3.20), we conclude that āii > 0 for i = 1, 2, . . . , n, which means
that (3.2) holds. By Theorem 3.1, system (1.1) has a positive ω-periodic solution
(x̃(t), ũ(t)). Let (x(t), u(t)) be any other positive solution of (1.1) with initial
condition (1.2). Consider the Liapunov functional V (t) = V (x̃(t), ũ(t)), (x(t), u(t)))
defined by
V (t) =
n
n X
m h
n
i
X
X
(1)
(2)
(3)
(4)
νi Vi (t) + Vi (t) +
Vijk (t) + Vijk (t)
i=1
m h
X
+
j=1 k=1
i
o
(5)
(6)
(7)
(8)
Vik (t) + Vik (t) + Vi (t) + Vi (t) ,
k=1
where, for i, j = 1, 2, . . . , n, k = 1, 2, . . . , m, we have
(1)
Vi (t)
Z
=
xi (t)
ds ,
gi (s)
x̃i (t)
t
Z
(3)
Vijk (t) =
Z
(4)
Vijk (t) =
t−τijk (t)
+∞ Z t
(7)
t
t−σik (s)
t
Z
Vik (t) =
Vi
t−γik (t)
+∞ Z t
Z
eik (ζik (s))
|ui (s) − ũi (s)| ds,
1 − σ̇ik (ζik (s))
pik (ξik (s))
|xi (s) − x̃i (s)| ds,
1 − γ̇ik (ξik (s))
fi (s + τ, s)|ui (s) − ũi (s)| ds dτ,
(t) =
0
(8)
Vi
bijk (µijk (s))
|xj (s) − x̃j (s)| ds,
1 − τ̇ijk (µijk (s))
t−τ
Vik (t) =
(6)
(t) = |ui (t) − ũi (t)|,
cijk (s + τ, s)|xj (s) − x̃j (s)| ds dτ,
0
Z
(5)
(2)
Vi
Z
t−τ
+∞ Z t
vi (s + τ, s)|ui (s) − ũi (s)| ds dτ
(t) =
0
t−τ
(3.21)
12
A. T. TRINH
EJDE-2013/261
Clearly V (t) is continuous on [0, +∞). Calculating the upper right derivative of
(8)
(t), . . . , Vi (t) along the solutions of system (1.1) for t > 0, we obtain
(1)
Vi
(1)
D + Vi
h ẋ (t)
x̃˙ i (t) i
i
=
sgn[xi (t) − x̃i (t)]
−
gi (xi (t)) gi (x̃i (t))
n
h X
= sgn[xi (t) − x̃i (t)] −
aij (xj (t) − x̃j (t))
j=1
−
−
−
n X
m
X
bijk (t)(xj (t
j=1 k=1
n X
m Z t
X
− τijk (t)) − x̃j (t − τijk (t)))
cijk (t, τ )(xj (τ ) − x̃j (t))dτ − di (t)(ui (t) − ũi (t))
j=1 k=1
m
X
−∞
Z
t
eik (t)(ui (t − σik (t)) − ũi (t − σik (t))) −
i
fi (t, τ )(ui (τ ) − ũi (τ ))dτ ;
−∞
k=1
and thus,
(1)
D + Vi
6 −aii (t)|xi (t) − x̃i (t)| +
n
X
aij (t)|xj (t) − x̃j (t)|
j6=i
+
+
+
m
n X
X
bijk (t)|xj (t
j=1 k=1
m Z t
n X
X
− τijk (t)) − x̃j (t − τijk (t))|
cijk (t, τ )|xj (τ ) − x̃j (τ )|dτ − di (t)|ui (t) − ũi (t)| (3.22)
j=1 k=1
m
X
−∞
eik (t)|ui (t − σik (t)) − ũi (t − σik (t))|
k=1
Z t
fi (t, τ )|ui (τ ) − ũi (τ )|dτ,
+
−∞
(2)
D+ Vi
= [u̇i (t) − ũ˙ i (t)] sgn[ui (t) − ũi (t)]
h
= sgn[ui (t) − ũi (t)] − αi (t)(ui (t) − ũi (t)) + βi (t)(xi (t) − x̃i (t))
+
m
X
pik (t)(xi (t − γik (t)) − x̃i (t − γik (t)))
k=1
Z t
vi (t, τ )(xi (τ ) − x̃i (τ ))dτ
+
i
−∞
≤ −αi (t)|ui (t) − ũi (t)| + βi (t)|xi (t) − x̃i (t)|
+
m
X
pik (t)|xi (t − γik (t)) − x̃i (t − γik (t))|
k=1
Z t
vi (t, τ )|xi (τ ) − x̃i (τ )|dτ ;
+
−∞
(3.23)
EJDE-2013/261
EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY
(3)
V̇ijk =
bijk (µijk (t))
|xj (t) − x̃j (t)|
1 − τ̇ijk (µijk (t)))
− bijk (t)|xj (t − τijk (t)) − x̃j (t − τijk (t))|,
(3.24)
+∞
Z
(4)
13
cijk (t + τ, t)|xj (t) − x̃j (t)|dτ
V̇ijk =
0
(3.25)
+∞
Z
−
cijk (t, t − τ )|xj (t − τ ) − x̃j (t − τ )|dτ ;
0
(5)
V̇ik
eik (ζik (t))
|ui (t) − ũi (t)| − eik (t)|ui (t − σik (t)) − ũi (t − σik (t))|;
=
1 − σ̇ik (ζik (t)))
(3.26)
(6)
V̇ik =
pik (ξik (t))
|xi (t) − x̃i (t)| − pik (t)|xi (t − γik (t)) − x̃i (t − γik (t))|;
1 − γ̇ik (ξik (t)))
(3.27)
Z +∞
(7)
fi (t + τ, t)|ui (t) − ũi (t)|dτ
V̇i =
0
(3.28)
Z +∞
−
fi (t, t − τ )|ui (t − τ ) − ũi (t − τ )|dτ ;
0
Z +∞
(8)
V̇i =
vi (t + τ, t)|xi (t) − x̃i (t)|dτ
0
(3.29)
Z +∞
−
vi (t, t − τ )|xi (t − τ ) − x̃i (t − τ )|dτ.
0
From (3.21)-(3.29) it follows that
D+ V ≤
n
m
nh
X
X
νi − aii (t) + βi (t) +
i=1
k=1
Z
+∞
+
pik (ξik (t))
1 − γ̇ik (ξik (t)))
i
vi (t + τ, t)dτ |xi (t) − x̃i (t)|
0
+
n
X
aij (t)|xj (t) − x̃j (t)| +
j=1 k=1
j6=i
+
+
=
m
n X
X
n X
m hZ
X
j=1 k=1
m
X
k=1
n n
X
+∞
n
i
o X
n
cijk (t + τ, t)dτ |xj (t) − x̃j (t)| +
νi − αi (t) + di (t)
0
i=1
eik (ζik (t))
+
1 − σ̇ik (ζik (t)))
+∞
Z
+
k=1
νj aji (t) +
j=1
j6=i
+
n
X
j=1
n
X
νj
m Z
X
k=1
0
+∞
νj
o
fi (t + τ, t)dτ |ui (t) − ũi (t)|
0
m
h
X
νi − aii (t) + βi (t) +
i=1
n
X
bijk (µijk (t))
|xj (t) − x̃j (t)|
1 − τ̇ijk (µijk (t)))
m
X
k=1
pik (ξik (t))
+
1 − γ̇ik (ξik (t))
bjik (µjik (t))
1 − τ̇jik (µjik (t))
o
cjik (t + τ, t)dτ |xi (t) − x̃i (t)|
Z
+∞
vi (t + τ, t)dτ
0
i
14
A. T. TRINH
EJDE-2013/261
n n h
m
X
X
νi − αi (t) + di (t) +
+
i=1
+∞
k=1
Z
fi (t + τ, t)dτ
+
io
eik (ζik (t))
1 − σ̇ik (ζik (t))
|ui (t) − ũi (t)|.
(3.30)
0
By (3.20), there exists positive number δ such that
min
t∈[0,ω]
−
n
X
m
n h
X
νi aii (t) − βi (t) −
k=1
νj aji (t) −
n
X
νj
j=1
min
t∈[0,ω]
νj
j=1
j6=i
−
n
X
m Z
X
m
X
k=1
pik (ξik (t))
−
1 − γ̇ik (ξik (t))
Z
+∞
vi (t + τ, t)dτ
i
0
bjik (µjik (t))
1 − τ̇jik (µjik (t))
(3.31)
+∞
cjik (t + τ, t)dτ
o
> δ,
0
k=1
m
n h
X
νi αi (t) − di (t) −
k=1
eik (ζik (t))
−
1 − σ̇ik (ζik (t))
Z
+∞
fi (t + τ, t)dτ
io
> δ,
0
for i = 1, 2, . . . , n. In the view of (3.30) and (3.31) we have
n n
o
X
D V (t) ≤ −δ
|xi (t) − x̃i (t)| + |ui (t) − ũi (t)| ,
+
t ≥ 0.
(3.32)
i=1
Integrating both sides of (3.32) from 0 to t, we obtain
Z tX
n n
o
V (t) − V (0) ≤ −δ
|xi (s) − x̃i (s)| + |ui (s) − ũi (s)| ds, t ≥ 0.
0 i=1
Thus,
Z tX
n n
o
V (0)
,
|xi (s) − x̃i (s)| + |ui (s) − ũi (s)| ds ≤
δ
0 i=1
t ≥ 0,
which implies
Z
n n
+∞ X
0
i=1
o
V (0)
.
|xi (s) − x̃i (s)| + |ui (s) − ũi (s)| ds ≤
δ
(3.33)
Since
n Z
νi xi (t)
x̃i (t)
o
ds + |ui (t) − ũi (t)| ≤ V (t) ≤ V (0),
gi (s)
t ≥ 0,
it follows from (H3) that xi (t) and ui (t) are bounded on [0, +∞), and hence from
(1.1) we can conclude that ẋi (t) and u̇i (t) are also bounded on [0, +∞).PThis implies
n
that xi (t) and ui (t) are uniformly continuous on [0, +∞). Therefore, i=1 |xi (t) −
x̃i (t)| + |ui (t) − ũi (t)| is uniformly continuous on [0, +∞). Thus, (3.33) implies that
lim
t→+∞
n n
o
X
|xi (t) − x̃i (t)| + |ui (t) − ũi (t)| = 0
i=1
and (x̃(t), ũ(t)) is the unique positive ω-periodic solution of system (1.1). The proof
is complete.
EJDE-2013/261
EXISTENCE AND GLOBAL ASYMPTOTIC STABILITY
15
As an example we consider system (1.1) with
n = 2,
ln 5
, α1 (t) = α2 (t) ≡ 6,
2
a12 (t) = a21 (t) = 1 + sin 2πt,
r1 (t) = r2 (t) ≡
m = 1,
a11 (t) = a22 (t) = 21 + sin 2πt,
b111 (t) = b121 (t) = b211 (t) = b221 (t) = 1 + 2 sin πt,
c111 (t, s) = c121 (t, s) = c211 (t, s) = c221 (t, s) = v1 (t, s) = v2 (t, s) = exp{−(t − s)},
d1 (t) = d2 (t) = 1 − sin 2πt,
e11 (t) = e21 (t) = 1 + cos 2πt,
β1 (t) = β2 (t) = 1 + cos 2πt,
p11 (t) = p21 (t) = 1 − cos 2πt,
τ111 (t) = τ121 (t) = τ211 (t) = τ221 (t) ≡ τ ∗ > 0,
σ11 (t) = σ21 (t) ≡ σ ∗ > 0,
γ11 (t) = γ21 (t) ≡ γ ∗ > 0, g1 (v) = g2 (v) = v.
It is easy to see that (H1)–(H5) hold. By straightforward computation, we have
c∗ij1 (t) = fi∗ (t) = vi∗ (t) = 1,
Ai = 24.8,
Bi = ln
ln 5
+ ln 5,
49.6
P̄i = Q̄i = R̄i =
ϕi (Bi ) =
5 ln 5
,
49.6
3
,
5
i, j = 1, 2.
Let ν1 = ν2 = 1. We can easily see that
Z
1
h
i
βi (s) + pi1 (s) + vi∗ (s) ds = 3 > 0,
0
2 X
j6=i
30 ln 5
ln 5
āij + b̄ij1 + c̄∗ij1 + P̄j + Q̄j + R̄j ϕj (Bj ) =
×
<
= r̄i ,
31
2
2
n h
min νi aii (t) − βi (t) −
t∈[0,2π]
−
2
X
j=1
νj
pi1 (ξi1 (t))
−
1 − γ̇i1 (ξi1 (t))
2
X
bji1 (µji1 (t))
−
νj
1 − τ̇ji1 (µji1 (t)) j=1
Z
Z
+∞
2
i X
vi (t + τ, t)dτ −
νj aji (t)
0
+∞
cji1 (t + τ, t)dτ
j6=i
o
0
= min [14 − cos 2π(t + γ ∗ ) − 2 sin 2π(t + τ ∗ )] > 0,
t∈[0,2π]
Z +∞
o
ei1 (ζi1 (t))
−
fi (t + τ, t)dτ
1 − σ̇i1 (ζi1 (t))
t∈[0,2π]
0
= min [3 + sin 2πt − cos 2π(t + σ ∗ )] > 0, i = 1, 2
min
n
αi (t) − di (t) −
t∈[0,2π]
Thuerefore, conditions (3.1), (3.3) and (3.30) hold. Hence, by Theorem 3.2, the
system has a unique positive 1-periodic solution which is globally asymptotically
stable.
Acknowledgements. The authors would like to thank the editors and the anonymous reviewers for their constructive comments. This work was supported by Hanoi
National University of Education and the Minister of Education and Training of
Vietnam.
16
A. T. TRINH
EJDE-2013/261
References
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Acta Mathematica Sinica, Vol. 19, 2003, pp. 801 - 822.
[2] Gains, R.E.; Mawhin, J.K.; Coincidence Degree and Nonlinear Differential Equations.
Springer-Verlag, Berlin. 1977.
[3] Liu, Z.J.; Tan, R.H.; Chen, L.S.; Global stability in a periodic delayed predator-prey system.
Applied Mathematics and Computation, Vol. 186, 2007, pp. 389 - 403.
[4] Yan, J.; Liu, G.; Periodicity and stability for a Lotka-Volterra type competition with feedback
controls and deviating arguments. Nonlinear Analysis, Vol. 74, 2011, pp. 2916 - 2928.
Anh Tuan Trinh
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy
Road, Hanoi, Vietnam
E-mail address: anhtt@hnue.edu.vn